The role of local structural disorder in non-Fermi liquid f-electron

intermetallics

Research funded in part by the Department of Energy, Basic Energy SciencesPresented at the workshop “Phase Competition in Transition-Metal Oxides and Other

Compounds” , to be held at the University of California at Berkeley on May 14-16, 2003

Corwin H. BoothChemical Sciences Division

Glenn T. Seaborg CenterLawrence Berkeley National

Laboratory

Outline

• Introduction and Motivation— Why worry or care about disorder?— How can we include disorder in the theor ies?— What can we do exper imentally?

• UCu5-xPdx (CeRhRuSi2)— a disordered system— disorder is par tially tunable— Main results are the same

• U3Ni3Sn4 C/T magnetic field dependence

• Conclusions

Disorder and hybr idization

non-Fermi liquidsUPdCu4

Anderson lattices

Yb1-xLuxAl3

Spin glassesURh2Ge2

(anti-) fer romagnets

UCu5

Hybridization →→ →→

Interaction disorder →→→→

NFL?

MAGNETIC GROUND STATES

U3Ni3Sn4

Teaser: Can molecular systems be included?

5 10 15 20 25 300

5000

10000

Inte

ract

ion

stre

ngth

V 2

RKKY Kondo T

N

binding energies: kBTRKKY~N(0) Vfd4/εεεεf

2

kBTK~exp[-εεεεf /N(0) Vfd2]

What happens when interactions are equal and drive a zero temperature

transition?

What if there is a distribution of interaction strengths?

NFL behavior in UCu5-xPdx

0

5

10

15

20

25

30

0 1 2 3 4 5

T(K

)

Pd concentration x

Cubic AuBe

5

Mixed Phase HexUPd

5

AFM

SG

NFL

TN

TK/10

TF

UCu5-x

Pdx

NFL behavior exists near phase boundaries

resistivity as T goes to 0 K has a weaker power law than T2

Susceptibility and heat capacityhave logarithmic divergences as T goes to 0 K.

Vollmer et al., PRB, 2000. Bernal et al. PRL, 1995. R. Chau, Ph.D. dissertation UCSD, 1998.

Unanswered questions

• Is disorder a necessary component?— first NFL’s were all substituted— many new “ ordered” ones coming online— even “ ordered” ones may have issues (eg. CeCu2Si2)

• Does disorder even matter?— well, how much are we talking about?

• Is this a new state of matter?

Three Possible Types of NFL Models(There are others: multichannel Kondo, etc…)

T

Tmag

x

NFL

FL

TK

P(TK) T

Disorder+Competition (Griffiths)NFL behavior due to proximity to a metal-insulator transition fixed point (Anderson localization,Miranda et al.) or to a magnetic/nonmagnetic fixed point (RKKY, Castro Neto et al.), each in presence of disorder and anisotropy.

Kondo Disorder ModelDisorder causes distribution of TK’s within a strict single-impurity model. Moments with TK< T are unquenched and give rise to NFL behavior.

Bernal et al. ‘95

T

Tmag

x

NFLFL

Quantum Critical PointNFL is generated from critical fluctuations above a zero-temperature critical point (Millis et al. ‘93) (Rappoport et al., 2001).

Effects of lattice disorder (Kondo lattice disorder model, or KLDM)

• Two types of lattice disorder : discrete and continuous

6

2/135 )(

df

dffdfd

R

rrV

−

= η ηηηηfd: constant rf: outer f-radiusrd: outer d-radius Rf-d: f-d bond length

tight-binding : model

2.0 2.2 2.4 2.6 2.8 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

P(R

) (1

/Å)

R (Å)=

bondsfdTotal VV

2Total

K )0(exp

VNTT f

F

ε−=Only Kondo:

interactions

TF: conduction band widthεεεεf: f-level energyVTotal: f/d hybr idization energy

• P(V) involves convolution with P(R)• rd var ies as species change

Harrison and Straub

NFL must have continuous disorder in KLDM!

0 100 200 300 400 5000.00

0.01

0.02

0.03(b)

J=7/2

χ(T)

(em

u/m

ol)

T (K)

0 100 200 300 400 5000.00

0.01

0.02

0.03

(a)

P(T K

) (K

-1)

no disorder σ=0.005 Å σ=0.010 Å σ=0.020 Å σ=0.040 Å

• For KLDM to work, must have weight in P(TK) at very low TK.

• Site interchange will never provide this weight by itself!

100 200 300 400 5000.00

0.05

0.10

0.15

0.20

binomial plus σ=0.001 Å plus σ=0.003 Å

P(T

K)

(K-1)

TK (K)

NFL behavior in UCu5-xPdx

• A distr ibution of TK’s can descr ibe all these data!

• Warning: this is pedantic: KDM has many problems!

Vollmer et al., PRB, 2000. Bernal et al. PRL, 1995. R. Chau, Ph.D. dissertation UCSD, 1998.

Inter ference of photoelectron waves

• Inter ference of outgoing and incoming part of photoelectron modulates absorption coefficient:

• Big advantage: Atomic-species specific.

• Disadvantages: very short range (<~5-6 Å), sensitive to multiple scatter ing, over lapping edges...

γ

functionon distributi-pair radial a is

)2sin()()(

))(1(0

2

g

drkrrgNk

k

irf

icii +∝

+=

⋅∝

φχ

χµµεµ

“ I was brought up to look at the atom as a nice hard fellow, red or grey in colour

according to taste.”

- Lord Rutherford

Extended x-ray absorption fine-structure (EXAFS)

17000 17200 17400 17600 17800 180000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

UCu4Pd

U LIII

edge

Nor

mal

ized

Abs

orpt

ion

E (eV)

• sample absorption is given by

µ t = loge(I1/I0)

• EXAFS χ(k)=[µ(k)-µ0(k)]/µ0(k)

-20

-10

0

10

20U L

I I I edge

k3 χ(

k)

k (Å-1)

1 2 3 4

-40

-20

0

20

40

16 U-Cu4 U-Pd12 U-Cu

r (Å)

FT o

f k3 χ(

k)

U L I I I edge

data fi t

Amplitude envelope

[Re2+Im2]1/2

Real part of the complex

transform

EXAFS

+∝i

cii drkrrkFrgNk )2sin(),()()( φχ

• U (4a) environment is identical to Pd (4c) environment, except U/Pd are switched. Nearest-neighbors are Cu (16e) at ~2.93 Å

• Cu environment differs due to tetrahedrons. Nearest-neighbors are Cu at 2.49 Å

• Determine amount of site interchange by number of Pd´-Cu pairs at 2.49 Å

• Definition: Pd´ denotes a Pd on a 16e site, Cu´ denotes a Cu on a 4c site.

3.06 Å

2.93 Å

2.93 Å

2.49 ÅLocal Copper environment

Local U and Pd environment

UCu4Pd average and local structurenominal

A “ zero-disorder” example: YbCu4X

0 1 2 3 4-60

-40

-20

0

20

40

60 F

T o

f k3 χ(

k)

r (Å)

Ag K edge YbCu4Ag

Pd K edge UCu4Pd

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

X-Cu

σ2 (Å

2 )

data fit X Tl In Cd Ag

0 50 100 150 200 250 3000.000

0.002

0.004

0.006

0.008

0.010

X-Yb

T (K)

σ2 (Å

2 )

X/Cu inter-

change

σstatic2(Å)ΘcD(K)

S02X Yb

2(2)%0.0006(5)0.0008(4)235(5)250(5)0.91(5)Ag

5(5)%0.0010(5)0.0007(4)255(5)240(5)0.98(5)Cd

2(3)%0.0011(5)0.0009(4)280(5)252(5)1.04(5)In

4(1)%0.0005(5)0.0004(4)230(5)230(5)0.89(5)Tl

YbCuCu

σσσσAB2(T)=σσσσstatic

2+F(µµµµAB,ΘΘΘΘcD)

J. L. Lawrence et al., PRB 63, 054427 (2000).

XAFS data on UCu5-xPdx

0 2 4 6 8 10 12 14

-0.2

0.0

0.2

-0.2

0.0

0.2

-0.3

0.0

0.3

0.6

(c)

Pd K edge

k (Å-1)

(b)

Cu K edge

k3 χ(k)

(a)

U LI I I

edge

0 1 2 3 4 5 6

-40

0

40

-40

0

40

-40

0

40

-40

0

40

(d)

UCu3.5

Pd1.5

FT o

f k3 χ(

k)

r (Å)

(c)

UCu4Pd

(b)

UCu4.3

Pd0.7

(a)

UCu5

U LIII edge

data fit

0 1 2 3 4 5 6

-40-20

02040

-40-20

02040

-40-20

02040

(d)

UCu3.5

Pd1.5

(c)

r (Å)

UCu4Pd

(b)

FT o

f k3 χ(

k)

UCu4.3

Pd0.7

(a)Pd K edge

-40-20

02040

UCu4.5

Pd0.5

data fit

3.06 Å

2.93 Å2.93 Å

2.49 Å

• Fit to all single-scatter ing paths out to the 16 Pd-Cu’s at ~4.59 Å.

• Including all site interchange, fits use 15 paths.

• Like bond lengths constrained together.

• Like bond length Debye-Wallers constrained together (σσσσA

2=(µµµµB/µµµµA)σσσσB2.

• Amplitude ratio’s constrained.Two possible descr iptions:s, x: s = NPd(16e)/NPd(Total)f4c

Pd, f16eCu: f4c

Pd is fraction of 4c sites with Pd, etc.

e.g. Pd´-Cu @ 2.5 Å has 6S02s f16e

Cu

neighbors

Pd K-edge fit results

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.00.10.20.30.40.50.60.70.80.91.0

0.0

0.1

0.2

0.3

0.4

0.5

(b)

Prob

abili

ty

x

f4c

Pd

f16e

Cu

nominal f4c

Pd

nominal f16e

Cu

(a)

s nominal s

C. H. Booth et al., PRL 81, 3960 (1998); E. D. Bauer et al., PRB 65, 245114 (2002).

No measurable continuous U-Cu disorder !

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

NFL from Kondo disorder

σ2 stat

ic f

or U

-Cu

(Å2 )

x

• NFL “ limit” for KLDM is generous: we estimate the best fit with 0.0034 Å2.

• Only the x=0.3 sample is anomalous… oxidation?

• Cu K edge fits indicate a nearest neighbor Cu-Cu distance of ~2.48 Å

• Pd K edge fits indicate a Pd´-Cu distance of ~2.55 Å

• Together no σσσσstatic2 for U-Cu, the

Cu displacements near a Pd´ must be near ly perpendicular to the U-Cu pairs.

2.93 Å

2.49 Å

Disorder : Is it enough?

• KDM: NFL is not from disorder in Vfd. This probably can’ t generate enough disorder in N(0) either (Miranda).

• RKKY clusters? ~0.5% of uranium environments have a Vfd that is equal to or less than that in UCu5 .

• Anderson localization? only 0.0025% of UCu5-like uraniums have a similar neighbor .

1 10 100

0.000

0.005

0.010

0.015

0.020

χ mag

(T)

(em

u/m

ol)

T (K)

UCu4Pd data

σstat

2=0.0

σstat

2=0.00020

σstat

2=0.00078

σstat

2=0.00137

σstat

2=0.00343

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

1

2

3

4

5

6

7

8

XAFS measure of U-Cu static disorder

KLDM fit to mag. suscept.

UCu5

P(V

)

V (eV)

σstat

2=0.0 Å2 (x50)

σstat

2=0.00020 Å2

σstat

2=0.00078 Å2

σstat

2=0.00137 Å2

σstat

2=0.00343 Å2

Effects of annealing

A. Weber et al., PRB 63, 205116 (2001).

• Annealing suppresses spin glass transition, removes linear resistivity but logar ithmic C/T remains ?!?!?

• Quick point: Entropy under this logar ithmic divergence is close to R log 2

0.1 1200

300

400

500

600

700

∆ C

/T (

mJ

mol

-1 K

-2)

T (K)

Unannealed 7 day 14 day

Both site interchange and bond length distr ibutions affected by annealing

1 2 3 4 50

10

20

30

40

50

60

700

10

20

30

40

U LI I I

edge(b)

FT M

agni

tude

of k

3 χ(k)

r (Å)

Pd K edge(a)

Unannealed 1 day 7 days 14 days

• Measure (discrete) site interchange with Pd K edge XAFS

• Measure (continuous) U-X bond length disorder with temperature dependence of distr ibution widths (Debye-Waller factors).

• Complication: U-Cu and U-Pd pairs strongly over lap, so need to be able to include degree of site interchange as a constraint to the U L I I I-edge fits.

• Solution: Fit to a site interchange model.

Effects of annealing

• Structurally, two things happen:— site interchange is reduced, but not

after more than 1 day of annealing— U-Cu bond length distr ibution

width decreases, even after 14 days of annealing

• Main points:— s decreases, but is still fair ly large— U-Cu orders, but it is already very

close to fully ordered (∆σ∆σ∆σ∆σ ~ -0.02 Å)

Unannealed 1-day 7-day 14-day

0.18

0.20

0.22

0.24

0.26

0.28

s (f

ract

ion

of P

d's

on 4

c si

tes)

Annealing time

0 3 6 9 12 15

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

∆σU

-Cu2 (Å

2 )

Annealing time (days)

“ Dark” width2KDM

2DM

2KDM

20

2

σ

+=

K

V

W

WWW

Effects of annealing

• f4cPd of unannealed samples very

consistent with changes in lattice parameter: x vs. f4c

Pd is linear , except with a change in slope at x ~ 0.85

• Annealing increases f4cPd, similar ly to

change in d• I t is possible to parameter ize changes in

heat capacity as ar ising only from changes in s and σσσσU-Cu

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.2

0.4

0.6

0.8

1.0

UCu5-x

Pdx

f 4c

Pd (

frac

tion

4c s

ites

with

Pd)

x

Unannealed from Bauer et al. Unannealed (this work) 1 day annealed (this work) nominal fit to x<0.8

0.1 1200

300

400

500

600

700

∆ C

/T (

mJ

mol

-1 K

-2)

T (K)

Unannealed 7 day 14 day

s=0.27, σKDM

2=0.00033 Å2

s=0.19, σKDM

2=0.00033 Å2

s=0.19, σKDM

2=0.00009 Å2

s=0.19, σKDM

2=0.0 Å2

• ANSWER(?): NFL state is somehow “ pre-loaded” , possibly as a consequence of disorder .

What the heck is W0?

• KDM by itself does not work!— linear resistivity goes away on annealing (Weber et al., PRB 63, 205116 (2001)

— µµµµSR indicates glassy spin dynamics (MacLaughlin et al., PRL 245114)— Shor t range (< unit cell) magnetic correlations exist (Aronson et al., PRL 87, 197205

(2001)

— Distr ibution of moments at high fields (>51 kOe) inconsistent with KDM (Buttgen et al., 62, 11545 (2000)

• Disorder can generate width in N(0) (not enough says Miranda, but could be says Cox)

• Is W0 due to a QCP? Idea is similar to proposed by Grempel and Rozenberg PRB 60, 4702 (1999), and to Rappoport et al., PRB 64, 140402 (2001).

• Is cluster ing important?

2Total

K )0(exp

VNTT f

F

ε−=

CeRhRuSi2

U3Ni3Sn4

Data summary

FL NFL

Susceptibility χχχχ(T) ∝∝∝∝ constant -log(T)Specific heat C(T)/T ∝∝∝∝ γγγγ -log(T)Electr ical resistivity ρρρρ(T) ∝∝∝∝ T2 T

G. R. Stewart, RMP, (2001)

C. Y. Liu et al., PRB 61, 432 (2000)

Non-Fermi liquid (NFL) and Ce(Ru1-xRhx)2Si2

Ce

Ru/Rh

Si

XAFS Study near Ce atom in CeRuRhSi2

Ce L 3-edgeT = 20K

Fitting parameters:S0² total data point var iables fitting freedom r -factor reduced-χχχχ²

0.85(10) 23 14 9 0.0054 3.26

Si Ru/Rh

Ce

Bonding length (Å):Ce-Si: 3.218(6)Ce-Ru/Rh: 3.254(3)Ce-Ce: 4.150(10)

Ce

Ru/Rh

Si

PDF Analysis of CeRuRhSi2

Total disorder factors:u11 (Ce) u11(Ru/Rh) u11 (Si)

0.000359(7) 0.0030(11) 0.0029(3)

Ce

Ru/Rh

Si

T = 20 K

Small total disorder factors suggest that static disorderis negligible!!!

Summary of structural disorder in CeRuRhSi2

σσσσ² (Ų) σσσσ²(static) (Ų) bonding length (Å)Ce-Si 0.0033(6) 0.0003(4) 3.215(6), 3.187Ce-Ru/Rh 0.0014(3) -0.0004(2) 3.254(3), 3.257Ce-Ce 0.0011(9) 0.0003(14) 4.15(1), 4.161Ru-Si 0.0011(3) -0.0003 2.373(3), 2.399Ru-Ru/Rh 0.0011(2) -0.0001 2.934(3), 2.888Ru-Ce 0.0022(2) -0.0003 3.231(4), 3.257Rh-Si 0.0015(3) 0.0004(3) 2.390(4), 2.399Rh-Ru/Rh 0.0012(1) -0.0003(2) 2.911(3), 2.888Rh-Ce 0.0025(3) -0.0002(2) 3.236(4), 3.257

u(11) (Ų)Ce 0.00036(1)Ru/Rh 0.0030(11)Si 0.0029(3)

XAFS:Short-range

PDF:Intermediate-range

T = 20 K

U3Ni3Sn4

Last words

Is U3Ni3Sn4 best descr ibed as near an AF QCP?

• U3Ni3Sn4 is an undoped, ambient pressure non-Fermi liquid.• Evidence of an AF cr itical point at –0.04 GPa (Estrela et al.,

(2001)). • A “ Her tz and Millis” Quantum Cr itical Point?

C/T ≈≈≈≈ γγγγ – A T0.5 () χ∝χ∝χ∝χ∝T-0.3 (?, 0.5) ∆ρ∝∆ρ∝∆ρ∝∆ρ∝T1.8 (?, 0.5)

• Cubic, bcc, I –43d, a0=9.3524 Å

• residual resistivity 7 µΩµΩµΩµΩ cm

• single crystal XRD good

• No temperature-dependent structural studies exist

• Disorder models have been shown to be capable of providing NFL behavior

No static offsets necessary

ΘΘΘΘcD(K)σσσσoffset2(Å2)Atom pair

159(4)-0.0007(3)U-U

241(1)-0.0006(3)U-Sn

252(5)-0.0009(4)U-Ni

No evidence of site interchange either…

Field-dependence of heat capacity

• Zero applied field, C/T~T0.5, indicative of NFL behavior

• Fermi liquid behavior appears to be recovered in relatively small applied fields

• (U3Ni3Sn4 behaves similar ly to CeCoIn5, another system with a “ negative pressure” cr itical point…)

0.04

0.06

0.08

0.10

1 2 3 4 5

0.04

0.06

0.08

0.10

1 2 3 4

0.04

0.06

0.08

0.10

B = 0 T

B = 8 T

B = 6 T

B = 4 T

C/T

(J/K

2 mol

eU)

B = 2 T

T 0.5 (K0.5)

B = 7 T

0 5 10 15

0.98

1.00

1.02

1.04

1.06

1.08

C(H

,T)/

C(0

,T)

T (K)

H=8 T H=7 T H=6 T H=4 T H=2 T

Compar ison to Gr ifftihs-McCoy… a Schottky anomaly?

2 4 6 8 10 12 140.9

1.0

1.1

1.2

1.3

C(8

T)/

C(0

T)

T (K)

data AF-GP ω

0=2.5 T+/-1.5 T,

q/γφ=0.04+/-0.01 AF-GP ω

0=100 T,

q/γφ=0.3TH

T

HTC /

2/3

2/2

eleffe/ µ

λ

λ−

−

+

∝

−−

∆∆+∆+∆∆∝

0

0

1

02222212el ln)(sech)(),(

ω θλ ωββ HH EEdTHC

HqE BH

φω

γµ

∆=∆ )ln(

1)( 0

• High-field limit:

µµµµeff is average effective moment of AF clusters… successfully applied to

La0.95Ce0.05RhIn5 (Kim et al, 2002)

• More generally:

∆∆∆∆ is the cluster tunneling energyωωωω0 is the tunneling energy for a single atom (cutoff)q average moment within a clusterγγγγ is an anisotropy parameter

Data summary

• UCu4Pd (Disordered NFL)— Pd/Cu site interchange, tunable by annealing— Very little bond-length disorder

• Not enough for KLDM (that’s all in Vfd)• Changes in annealing indicate there is at least a little, and it does

affect the magnetic properties

• CeRhRuSi2 (Disordered NFL)— Very little, if any, bond length disorder

• annealing has not, thus far , produced any change in any properties

• U3Ni3Sn4 (Ordered NFL)— Very little, if any, structural disorder

• CeRhIn5, CeI r In5, Ce2RhIn8, Ce2I r In8 (Ordered NFL’s)— Very little, if any structural disorder

Last words

• Nature is sneaky: lattice disorder can hide!• For UCu4Pd, KLDM (Kondo lattice disorder

model), with no disorder in N(0) is not enough.• Role of disorder still very much unclear !

— Does disorder even matter? Yes, but it can’ t explain everything!

— definitely not conventional: either extremely sensitive or it is a minor player

• Cluster ing? Magnetic droplets? Gr iffiths-McCoy?— Probably not exactly Griffiths-McCoy— Could be… tough to see structurally

• Should doped and undoped systems be treated in the same way?— I ’m leaning toward yes…

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

1

2

3

4

5

6

7

8

XAFS measure of U-Cu static disorder

KLDM fit to mag. suscept.

UCu5

P(V

)

V (eV)

σstat

2=0.0 Å2 (x50)

σstat

2=0.00020 Å2

σstat

2=0.00078 Å2

σstat

2=0.00137 Å2

σstat

2=0.00343 Å2

0

5

10

15

20

25

30

0 1 2 3 4 5

T(K

)

Pd concentration x

Cubic AuBe

5

Mixed Phase HexUPd

5

AFM

SG

NFL

TN

TK/10

TF

UCu5-x

Pdx

0.1 1200

300

400

500

600

700

∆ C

/T (m

J m

ol-1 K

-2)

T (K)

Unannealed 7 day 14 day

s=0.27, σKDM

2=0.00033 Å2

s=0.19, σKDM

2=0.00033 Å2

s=0.19, σKDM

2=0.00009 Å2

s=0.19, σKDM

2=0.0 Å2

Last words (continued…)

• Competing interaction descr iptions seem most appropr iate. Is RKKY or Anderson localization the important competing interaction?Structurally, system seems to cross to RKKY, but is close to the boundary!

• KDM could still work, if something else amplifies the effect of the disorder (“ pre-loading” ).

• Even the “ canonical” disordered NFL CeRhRuSi2is remarkably well ordered

• Should the disordered and the “ well ordered” NFL’s be considered as a whole?

• U3Ni3Sn4 passes all the tests of a well ordered NFL

• In addition, FL/NFL development in field appears to not be the product of a Gr iffiths-McCoy singular ity

• (U3Ni3Sn4 behaves similar ly to CeCoIn5, another system with a “ negative pressure” cr itical point…)

2 4 6 8 10 12 140.9

1.0

1.1

1.2

1.3

C(8

T)/

C(0

T)

T (K)

data AF-GP ω

0=2.5 T+/-1.5 T,

q/γφ=0.04+/-0.01 AF-GP ω

0=100 T,

q/γφ=0.3

Collaborators and Acknowledgements

S.-W. Han (LBNL)E. D. Bauer , R. Chau, M. B. Maple (UC San Diego)E.-W. Scheidt, S. Kehrein, A. Weber , U. K iller (Universität Augsburg)J. L . Sarrao (Los Alamos National Laboratory)L . E. De Long, J. G. Huber (University of Kentucky)L. Shlyk, K. Nenkov (IFW, Dresden)G. H. Kwei (Lawrence Livermore National Laboratory)

Thanks to Jon Lawrence, Frank Br idges, Andrew Cornelius, Greg Stewar t, Antonio Castro Neto, Dan Cox, Joe Thompson and Mike Hundley for discussions and assistance.

Thanks to the Office of Basic Energy Science for funding part of this work.Data were collected on Beamlines 4-3, 10-2 and 11-2 at the Stanford

Synchrotron Radiation Laboratory (SSRL), which is operated by the DOE, OBES.